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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2017, Vol. 34 ›› Issue (2): 199-208.doi: 10.3969/j.issn.1005-3085.2017.02.008

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一类非凸集值优化问题的对偶定理（英）

余国林,   马小军   

  1. 北方民族大学应用数学研究所，银川  750021
  • 收稿日期:2015-03-31 接受日期:2016-11-13 出版日期:2017-04-15 发布日期:2017-06-15
  • 基金资助:
    国家自然科学基金 (11361001).

Duality Theorems for a Nonconvex Set-valued Optimization Problem

YU Guo-lin,  MA Xiao-jun   

  1. Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan 750021
  • Received:2015-03-31 Accepted:2016-11-13 Online:2017-04-15 Published:2017-06-15
  • Supported by:
    The National Natural Science Foundation of China (11361001).

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摘要：

对偶理论是数学规划研究领域的重点问题之一，通对偶模型可以实现一个最小化问题与一个最大化问题之间的相互转化．本文的目的是建立一类非凸约束集值优化问题的对偶理论，在逼近多值函数定义的不变凸性假设下，研究了原集值优化问题的Mond-Weir型和Wolfe型对偶问题．利用分析的方法，本文得到了两种对偶模型下关于弱极小元的弱对偶定理，强对偶定理和逆对偶定理．这些对偶定理揭示了原问题与所讨论的Mond-Weir型和Wolfe型对偶问题之间存在着明确的对偶关系．本文所得结果丰富和深化了集值优化理论及其应用的研究内容．

关键词: 集值优化, 凸分析, 最优性条件, 多值函数逼近, 对偶

Abstract:

Duality is of great importance in mathematical programming, since it allows to study a minimization problem through a maximization problem and to know what one can expect in the best case and has resulted in many applications. The aim of this paper is to establish the duality theorems for a kind of nonconvex constraint set-valued optimization problems. Based on the notion of invexity in terms of cone-approximating multifunction for a set-valued map, Mond-Weir and Wolfe dual problems are investigated for a primal constraint set-valued optimization. By employing the analytic method, the weak duality theorems, the strong theorems and the converse duality theorems between Mond-Weir and Wolfe dual problems and the primal constraint set-valued optimization problem are established in sense of weak efficiency. These duality theorems disclose that there exist the precise dual relationships between the primal optimization and the involved dual problems. The results obtained in present paper enrich and deepen the theory and applications of set-valued optimization.

Key words: set-valued optimization, convex analysis, optimality conditions, approximating multifunction, duality

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